Parallel Second-Order Expansion

The general parallel second-order model can be written

$$H(z)=\sum_{k=1}^N \frac{b_k + c_k z^{-1}}{1 - 2 r_k \cos(\theta_k) z^{-1} + r_k^2 z^{-2}},$$

where typically $r_k = \exp(-\pi b_k T)$, $\theta_k = 2\pi f_k T$, and $T$ is the sampling period in seconds. The numerator coefficients are related to the fundamental mode parameters by

$$\begin{eqnarray*} b_k & = & 2 a_k \cos(\phi_k) \nonumber \\ c_k & = & -2 a_k r_k \cos(\phi_k - \theta_k) \nonumber \end{eqnarray*}$$

While the direct modal expansion technique is fully general, it does not take advantage of structure in the modal tunings common in musical instruments.